Bayesian Updating and Evidence Assessment

Bayes' theorem in odds form says posterior odds = prior odds × likelihood ratio, where the likelihood ratio LR = P(E | H) / P(E | not-H). Each independent piece of evidence multiplies the odds by its own LR. An LR above 1 favours H; below 1 favours not-H; equal to 1 is uninformative. Because updates multiply, several moderately diagnostic pieces of independent evidence can outweigh one dramatic but weakly diagnostic one, and correlated evidence (several studies sharing one flawed dataset) should be counted closer to once.

Judges and debaters do something like this informally. A good speaker explains not only that evidence supports their claim but how strongly, by asking how likely the evidence would be if their opponent were right. Evidence that both sides predict (for example, “some redevelopment projects succeeded”) has an LR near 1 and should not move the judge. Evidence the other side cannot easily explain (for example, a well-designed comparison showing displacement was concentrated among renters) has a high LR. Bayesian thinking also explains why burden matters: a claim with a low prior, such as a policy producing a dramatic change, needs stronger evidence.

In the scientific-reproducibility motion, suppose your prior that a particular published effect is real is 50%. A large replication project finds that a comparable class of findings replicated less often than expected: treat it as evidence with LR 0.6 against this kind of result, lowering odds from 1 to 0.6 (37.5%). A pre-registered direct replication then succeeds (LR 4): odds become 2.4 (about 71%). A press release repeating the original claim adds nothing (LR ≈ 1), because it would exist whether or not the effect is real. Use the interactive ledger to build your own sequence.

The likelihood ratios in debates are judgments, not measurements, and priors differ between people. Bayesian updating also applies to factual beliefs, not to the normative question of which value should take priority. It formalises consistency; it does not guarantee that the inputs are right.